# Singularities in black holes

I've read that a singularity is formed when the increasing gravity of a collapsing star overcomes its increasing density, resulting in an infinitely dense "point" of zero dimensions. But the mathematics seems to say the opposite.

Newton's famous formula for gravitational attraction has d squared in the denominator, where d in this case is the radius of the collapsing star. The formula for density in this case has the radius cubed in the denominator, as the star's volume is a function of the cube of its radius. Thus, as these two denominators shrink as the star collapses, its density grows as the cube of the radius but the gravitational force grows only as the square of the radius.

So how does the star's increasing gravity overcome its more rapidly increasing density?

• In GR, unlike Newtonian gravity, pressure is also a gravitational source term. See, for example, this - "Moreover, above a mass of about 2 solar masses a nonrotating neutron star will inevitably collapse to form a black hole, thanks in part to the gravitational attraction caused by pressure." Oct 26 '17 at 22:15
• @AlfredCentauri That sounds like it should be an answer Oct 27 '17 at 0:30
• one should also keep in mind that quantization of gravity has to enter before the classical singularty, and quantization does not allow singularities, so it is a problem that will be completely solved once gravity is definitively quantzed. Oct 27 '17 at 3:29
• Hi. You can see this from the relativistic TOV equation, see this related answer: physics.stackexchange.com/a/341102/28370 Oct 27 '17 at 3:30
• anna v: Do you mean the discovery of gravitons? Oct 28 '17 at 14:24

But black holes 'live' in the context of General Relativity, Einstein's theory of Gravitation, rather than Newtonian gravity.

In GR, unlike Newtonian gravity, pressure is also a gravitational source term and so, as the star shrinks beyond a certain point, the increasing pressure actually promotes, rather than resists, further contraction.

See, for example, Gravitational Collapse where we find:

Moreover, above a mass of about 2 solar masses a nonrotating neutron star will inevitably collapse to form a black hole, thanks in part to the gravitational attraction caused by pressure.

• Alfred: Many thanks for your response to my query about how singularities are formed, specifically the tug-of-war between density and gravity. Pressure is new to me, although I've done some reading about SR and GR. I'm starting Kip Thorne's book, "Black Holes and Time Warps" and hope to learn more about pressure. Oct 28 '17 at 0:46

The gravity has to overcome pressure not density (though pressure does increase with density, just not as fast as gravity, once gravity becomes strong enough) The handy, if emotional, mnemonic is "Gravity sucks, and pressure is repulsive." All real matter has pressure, ideal gas being the one example studied in high-school or freshman physics classes. There pressure is indeed the cube of size, and gravity is quadratic in size for the same mass, so pressure wins. However, as things get compressed enough, the particles the gas is composed of start moving close to the speed of light, limiting how fast the pressure can grow. Eventually gravity starts growing faster, and the game is over. Well, not quite over, first the regular star collapses into a white dwarf, then into a neutron start (and then potentially into a quark star, that part is not 100% clear), and after each step the pressure has more room to grow roughly cubically, for a little while, but then gravity wins again. And once the matter is compressed enough, there is no coming back yet again.

• SergeS: Many thanks for your response to my query about how a singularity is formed, specifically the tug-of-war between density and gravity. Oct 28 '17 at 0:37
• SergeS: I pressed Return to start a new paragraph not knowing it would send my unfinished comment. (I'll now try Shift/Return.) (Worked!) Pressure is a new concept to me, although I've done a lot of reading on SR and GR. I'm starting Kip Thorne's book "Black Holes and Time Warps" and hope to learn more about pressure. Oct 28 '17 at 0:42